The Algorithmic Phase Transition in Correlated Spiked Models

We study the computational task of detecting and estimating correlated signals in a pair of spiked matrices $X=\tfrac{\lambda}{\sqrt{n}} xu^{\top}+W, \quad Y=\tfrac{\mu}{\sqrt{n}} yv^{\top}+Z$ where the spikes $x,y$ have correlation $\rho$. Specifically, we consider two fundamental models: (1) Correlated spiked Wigner model with signal-to-noise ratio $\lambda,\mu$; (2) Correlated spiked $n*N$ Wishart (covariance) model with signal-to-noise ratio $\sqrt\lambda,\sqrt\mu$.